Compact composition operators on Hardy-Orlicz and Bergman-Orlicz spaces

arXiv:0910.5368v2 [math.FA] 31 May 2010 Compact composition operators on Bergman-Orlicz spaces Pascal Lefèvre, Daniel Li, Hervé Queffélec, Luis Rodríguez-Piazza June 1, 2010 Abstract. We construct an analytic self-map ϕof the unit disk and an Orlicz function Ψfor which the composition operator of symbol ϕis compact on the Hardy-Orlicz space HΨ, but not on the Bergman-Orlicz space BΨ. For that, we first prove a Carleson embedding theorem, and then characterize the compactness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order 2). We show that this Carleson function is equivalent to the Nevanlinna counting function of order 2. Mathematics Subject Classification. Primary: 47B33 – Secondary: 30D50; 30D55; 46E15 Key-words. Bergman-Orlicz space – Carleson function – Compactness – Composition operator – Hardy-Orlicz space – Nevanlinna counting function 1 Introduction and notation 1.1 Introduction Due to the Littlewood subordination principle, the boundedness of composition operators Cϕ, defined by Cϕ(f) = f◦ϕ, on Hardy spaces Hp, as well as on Bergman spaces Bp,1≤p≤ ∞, is automatic. Their compactness is something much more subtle, but is well understood now, and there are two well-separated cases. First, the case p=∞, for which Cϕ:H∞→H∞is compact if and only if kϕk∞<1(note that B∞=H∞). Secondly, the case p < ∞, for which the compactness does not depend on p. For Hardy spaces, this fact, proved by J. Shapiro and P. Taylor ([16]), is not completely trivial, and is due to the good factorization properties of functions in Hp. For the scale of Bergman spaces Bp, the factorization properties are not so good, but the independence with respect to pfollows from the following characterization ([12], Corollary 4.4): for 1≤p < ∞,Cϕ:Bp→Bpis compact if and only if the pull-back measure of the area-measure Aby ϕis a vanishing 2-Carleson measure. The case p= 2 1 (proved in [1]) gives then, for 1≤p < ∞: (1.1) Cϕ:Bp→Bpis compact ⇐⇒ lim |z|→1 1−|ϕ(z)| 1−|z|=∞. In both cases (Hardy and Bergman), a brutal change of situation occurs when we pass from finite values of pto the value p=∞, and the need was felt for an intermediate scale between Hpand H∞, or between Bpand B∞. This is what we did ([7]), in full detail, with Hardy-Orlicz spaces HΨassociated with an Orlicz function Ψ(and began to do for Bergman-Orlicz spaces BΨ). We introduced a generalization of the notion of Carleson measure, and proved a contractivity property of those Carleson measures mϕ=ϕ∗(m), attached to an analytic self-map ϕ:D→D, which turned out to be central to obtain a necessary and sufficient condition for the compactness of Cϕ:HΨ→HΨ. In that paper, we also began a study of the compactness of composition operators on BΨ. We proved, in particular, but implicitly (see the comments at the beginning of Section 4), that, if the Orlicz function Ψgrows very fast (satisfying the so-called ∆2condition), then the compactness of Cϕ:HΨ→HΨimplies its compactness as an operator Cϕ:BΨ→BΨ. On the other hand, it is well-known that the compactness on Hpimplies the compactness on Bpbecause it is easy to see that the right-hand side of (1.1) is implied by the compactness on Hp. One might think that it is generally easier to achieve compactness on BΨthan on HΨ. The main result of the present work is the existence of an analytic self-map ϕof D and an Orlicz function Ψsuch that the composition operator Cϕis compact on HΨbut not on BΨ. For that, we first have to characterize the compactness of composition operators on Bergman-Orlicz spaces. More precisely, the paper is organized as follows. In Section 2, given two Orlicz functions Ψ1and Ψ2, and a finite positive measure µon the unit disk D, we investigate under which conditions the canonical inclusion Iµ:BΨ1→LΨ2(µ), defined by Iµ(f) = f, is either bounded, or compact. In Theorem 2.1, we give a necessary condition and a sufficient condition, in terms of the Carleson function ρµof µ, for the boundedness of Iµ. Analoguously, we have a similar statement (Theorem 2.5) for the compactness of Iµ. In general, these necessary and those sufficient conditions do not fit. In Section 3, we prove one of the main results of this paper (Theorem 3.1) under the form of a contractivity principle for the pull-back measure Aϕof the planar Lebesgue measure Aon Dby ϕ. The proof is rather long and uses a Calderón-Zygmund decomposition, as well as an elementary, but very useful, inequality due to Paley and Zygmund. This contractivity principle eliminates the absence of fitness mentioned above and allows us to have a necessary and sufficient condition for the compactness of Cϕ:BΨ→BΨin terms of the same Carleson function ρAϕ=ρϕ,2(Theorem 3.2). In Subsection 3.2, we consider the Nevanlinna counting function Nϕ,2(initiated in [15]), adapted to the Bergman case, and we compare it with the 2Carleson function ρϕ,2of ϕ. These two functions turn out to be equivalent, in the sense precised in Theorem 3.10. This extends to the Bergman case (and fol2 lows from) such an equivalence for the Hardy case, that we recently established in [9], Theorem 1.1. Finally, in Section 4, we exploit the necessary and sufficient conditions that we established, either on HΨand on BΨ, to give (Theorem 4.2) an example of an analytic self-map ϕ:D→Dand of a fairly irregularly varying Orlicz function Ψ such that, contrary to the general intuition, Cϕ:HΨ→HΨis compact, whereas Cϕ:BΨ→BΨis not compact. This is due to the fact that we can evaluate, in an accurate way, the two Carleson functions ρϕand ρϕ,2of ϕ. Acknowledgement. The fourth-named author is partially supported by a Spanish research project MTM2006-05622. 1.2 Notation We shall denote by Dthe open unit disk {z∈C;|z|<1}of the complex plane, and its boundary, the unit circle, by T. The normalized area measure dA=dx dy/π on Dwill be denoted by A. For any ξ∈T, we define, for 0< h < 1, the Carleson window W(ξ, h)by W(ξ, h) = {z∈D;|z| ≥ 1−hand |arg(zξ)| ≤ πh}. We shall also use the “circular” Carleson windows S(ξ, h)defined by S(ξ, h) = {z∈D;|z−ξ|< h}. Since S(ξ, h)⊆W(ξ, h)⊆S(ξ, 5h), the measures of W(ξ, h)and of S(ξ, h)are equivalent, up to constants. For any finite positive measure µon D, we define, for 0< h ≤1, the Carleson function of µby: (1.2) ρµ(h) = sup |ξ|=1 µW(ξ, h), and we set: (1.3) Kµ,2(h) = sup 0<t<h ρµ(t) t2 When ρµ(h) = O(h2), one says that µis a 2-Carleson measure; we also say that µis a Bergman-Carleson measure, to insist that the order 2is adapted to the Bergman spaces. When µ=Aϕis the pull-back measure of Aby an analytic self-map ϕ:D→D, we shall simply write ρϕ,2and Kϕ,2instead of ρAϕand KAϕ,2respectively. We shall say that ρϕ,2the 2-dimensional Carleson function of ϕ. The Hastings-Luecking sets of size 2−nare defined by: ∆k=nz∈D; 1 −1 2n≤ |z|<1−1 2n+1 and (2j−1)π 2n≤arg z < (2j+ 1)π 2no, where k= 2n+j−1,n≥0,0≤j≤2n−1(note that ∆0=D(0,1/2)). 3 An Orlicz function Ψis a positive increasing convex function Ψ: [0,∞)→ [0,∞)such that Ψ(0) = 0 (and Ψ(∞) = ∞). If µis a positive measure on D, the Orlicz space LΨ(µ)is the space of (classes of) measurable functions f:D→C such that RDΨ(|f|/C)dA<∞, for some constant C > 0, and the norm kfkΨ is defined as the infimum of all constants C > 0for which RDΨ(|f|/C)dA ≤ 1. The Bergman-Orlicz space is the subspace of LΨ(A)whose members are analytic in D. The Hardy-Orlicz space HΨis the subspace of H1whose boundary values are in the Orlicz space LΨ(T, m). We refer to [3] (see also [5], and [17]) for the theory of Bergman spaces and to [14] for more information about Orlicz spaces. 2 Carleson embeddings We consider in this Section the “embedding” map Iµ:BΨ1→LΨ2(µ), defined by Iµ(f) = f, where µis an arbitrary finite positive Borel measure on D and Ψ1and Ψ2are two Orlicz functions. 2.1 Boundedness Theorem 2.1 Given µa finite positive Borel measure on Dand Ψ1and Ψ2 two Orlicz functions, let Iµ:BΨ1→LΨ2(µ)be the canonical map defined by Iµ(f) = f. One has: 1) If Iµis bounded, then there is a constant A > 0such that: (2.1) ρµ(h)≤1 Ψ2[AΨ−1 1(1/h2)] ,for all 0< h < 1. 2) In order that Iµis bounded, it suffices that there is a constant A > 0such that: (2.2) Kµ,2(h)≤1/h2 Ψ2[AΨ−1 1(1/h2)] ,for all 0< h < 1. Note that condition (2.1) reads as Ψ−1 1(1/h2) Ψ−1 21/ρµ(h)is bounded (by 1/A) and condition (2.2) as Ψ−1 1(1/h2) Ψ−1 21/h2Kµ,2(h)is bounded. When Ψ1= Ψ2= Ψ and the Orlicz function Ψsatisfies the usual condition ∆2:Ψ(2x)≤CΨ(x)for some constant C > 1and xlarge enough, it is clear that conditions (2.1) and (2.2) are equivalent. However, they are not equivalent in general; and even condition (2.1) is not sufficient and condition (2.2) is not necessary: the examples 1.b and 2. of [7], Chapter 4, §3, given in the Hardy case, work also for the Bergman case. For the sake of completeness, we are going to sketch them. 4 Example 1. For every Orlicz function which does not satisfy the ∆2condition, there exists a finite positive measure µon Dsuch that Iµ:BΨ→LΨ(µ)is continuous, though µis not a 2-Carleson measure, and a fortiori does not verify (2.2). Proof. Since Ψdoes not satisfy ∆2, there exists an increasing sequence (an)n≥1 such that Ψ(2an)/n is increasing and Ψ(2an)/Ψ(an)≥n2n. Define the discrete measure µ: µ=∞ X n=1 n Ψ(2an)−n+ 1 Ψ(2an+1)δxn, where xn= 1 −1/pΨ(2an). As µ[xN,1]=N/Ψ(2aN),µis not a 2-Carleson measure. On the other hand, for every fin the unit ball of BΨ, one has ([7], Lemma 5.2) |f(z)| ≤ 8 Ψ−1[1/(1 − |z|)2]and it is easy to check that, if g(z) = Ψ−1[1/(1 −|z|)2], then kgkLΨ(µ)≤2, so kfkLΨ(µ)≤16, proving that Iµ is bounded.  Example 2. Let Ψ(x) = ex−1; there exists a finite positive measure µon D such that (2.1) holds but Iµ:BΨ→LΨ(µ)is not bounded. Proof. Let νbe a probability measure on T, supported by a compact set L of Lebesgue measure zero, such that ν(I)≤ |I|1/2, for each interval I. We can associate to νthe measure on Ddefined by ˜ν(E) = ν(E∩T). By RudinCarleson’s Theorem, for every integer n, there exists a function gnin the unit ball of the disk algebra such that |gn|= 1 on Land kgnkHΨ≤4−n. As Lis compact, there exists some rn∈(1/2,1) such that |gn(rnz)| ≥ 1/2for every z∈L. Now, define the measure µby: µ(E) = ∞ X n=1 1 2nνn(E), where: νn(E) = ν{z∈T;rnz∈E}. If Wis a Carleson window of size hthen, for each n≥1, we have: ν{z∈T;rnz∈W}≤ν(W∩T)≤(2h)1/2. Hence, µ(W)≤(2h)1/2.1/Ψ[1 4Ψ−1(1/h2)], and the condition (2.1) is fulfilled. Nevertheless, the identity from BΨto L1(µ)is not continuous since this would imply that the identity from HΨto L1(µ)were continuous as well, which is not the case: kgnkL1(µ)≥1/2n+1. In order to prove Theorem 2.1, we shall need some results. They are analogous to Proposition 4.9, Theorem 4.13 and Lemma 4.14 of [7], but their proofs require different arguments1. 1By the way, we seize the opportunity to correct here the proof of Theorem 4.13 given in [7], where some argument had been put awkwardly. In that proof, we first had to set M={z∈D;|z|>1−hand |f(z)|> t}. Then, Mfbeing the non-tangential maximal function of f∈H1, the open set {Mf> t}is the disjoint union of a countable family of open arcs Ij⊆T, and we had to say that every zsuch that |f(z)|> t belongs to some window W(Ij)(see [2], page 39). 5 We first introduce the following maximal function: (2.3) Λf=∞ X k=0 sup ∆k|f|1I∆k. One has: Lemma 2.2 For every Orlicz function Ψ, the map f∈BΨ7→ Λf∈LΨ(D)is bounded. Proof. Fix f∈BΨ. Set ck= sup∆k|f|for every k≥1, and let αk∈∆kbe such that |f(αk)| ≥ 1 2sup∆k|f|=ck/2. With C=kfkBΨ>0, one has: ZD Ψ(Λf/2C)dA=X k≥0 Ψ(ck/2C)A(∆k)≤ZD Ψ(|f|/C)dµ , where µ=Pk≥0A(∆k)δαk. But, for every Carleson window W, we can write: µ(W) = X αk∈WA(∆k)≤X ∆k∩W6=∅A(∆k) = A[ ∆k∩W6=∅ ∆k, and, since S∆k∩W6=∅∆kis contained in the window ˜ Wwith the same center as W, but with size two times that of W, one has µ(W)≤ A(˜ W) = 4 A(W). Hence µis a Bergman-Carleson measure. By [4], it follows that, for some constant C0>0, which does not depend on f, one has, using the subharmonicity of Ψ(|f|/C): ZD Ψ(|f|/C)dµ ≤C0ZD Ψ(|f|/C)dA ≤ C0. We shall, as we may, assume that C0≥1. Now, by convexity of Ψ, we get: ZD ΨΛf 2C0kfkBΨdA ≤ ZD 1 C0 ΨΛf 2kfkBΨdA ≤ 1, meaning that kΛfkLΨ(D)≤2C0kfkBΨ. Lemma 2.3 For every f∈B1and every finite positive Borel measure µon D, one has, for 0< h < 1/2and t > 0: µ({z∈D;|z|>1−hand |f(z)|> t})≤4Kµ,2(2h)A({Λf> t}). Proof. Remark that when z∈∆kand |z|>1−h, we must have 1−2−n−1> |z|>1−h, hence h > 2−n−1; since k= 2n+j−1≥2n−1, we must have k≥1/4h. Let I={k≥1 ; sup∆k|f|> t}and Ih={k≥1/4h; sup∆k|f|> t}. 6 If Wkis the smallest Carleson window containing ∆k, we have: µ({z∈D;|z|>1−hand |f(z)|> t}) ≤X k∈Ih µ(∆k)≤X k∈Ih, k≥1/4h µ(Wk) .X k∈Ih Kµ,2(2h)A(Wk)≤4X k∈Ih Kµ,2(2h)A(∆k) ≤4Kµ,2(2h)X k∈IA(∆k) = 4Kµ,2(2h)A({Λf> t}). and Lemma 2.3 is proved.  Lemma 2.4 Let µbe a finite Borel measure on Dand Ψ1and Ψ2two Orlicz functions. We suppose that, for some positive constant A, there is 0< hA≤1/2such that Kµ,2(h)≤1/h2 Ψ2[AΨ−1 1(1/h2)] ,for 0< h < hA. Then, for every f∈BΨ1such that kfkBΨ1≤1and every Borel subset Eof D, one has, with xA= (A/8)Ψ−1 1(4/h2 A): ZE Ψ2(A|f|/64) dµ ≤µ(E)Ψ2(xA) + 1 8ZD Ψ1(Λf)dA. Proof. For every s > 0, the inequality |f(z)|> s implies that the norm of the evaluation δzat zis greater than s. But this norm is ([7], Lemma 5.2) Ψ11/(1 −|z|)2, up to constants; more precisely: kδzk ≤ 8 Ψ−1 11 (1 −|z|)2· Hence, we have: s < 8 Ψ−1 11 (1 −|z|)2, so: |z|>1−1 pΨ1(s/8) · Lemma 2.3 gives, when Ψ1(s/8) ≥2: µ({|f(z)|> s}) = µ({|z|>1−1 pΨ1(s/8) and |f(z)|> s} ≤4Kµ,22 pΨ1(s/8)A({Λf> s}). 7 But, by our assumption, if Ψ1(s/8) ≥4/h2 A, Kµ,22 pΨ1(s/8)≤Ψ1(s/8)/4 Ψ2[AΨ−1 1(Ψ1(s/8)/4] ≤Ψ1(s/8)/4 Ψ2[(A/4)Ψ−1 1(Ψ1(s/8)](by convexity) =1 4 Ψ1(s/8) Ψ2(As/32) ; hence: µ({|f(z)|> s})≤Ψ1(s/8) Ψ2(As/32) A({Λf> s}). We get therefore: ZE Ψ2(A|f|/64) dµ =Z+∞ 0 Ψ′ 2(t)µ({|f|>64t/A}∩E)dt ≤ZxA 0 Ψ′ 2(t)µ(E)dt +Z+∞ xA Ψ′ 2(t)Ψ1(8t/A) Ψ2(2t)A({Λf>64t/A})dt ≤Ψ2(xA)µ(E) +Z+∞ xA Ψ′ 2(t) Ψ2(2t)Ψ1(8t/A)A({Λf>64t/A})dt . But, as Ψ1and Ψ2are Orlicz functions, one has tΨ′ 2(t)≤Ψ2(2t)and Ψ1(8t/A)≤(8t/A)Ψ′ 1(8t/A); hence: Z+∞ xA Ψ′ 2(t) Ψ2(2t)Ψ1(8t/A)A({Λf>64t/A})dt ≤Z+∞ 0 Ψ1(8t/A) tA({Λf>64t/A})dt ≤8 AZ+∞ 0 Ψ′ 1(8t/A)A({Λf>64t/A})dt =Z+∞ 0 Ψ′ 1(x)A({Λf>8x})dx =ZD Ψ1(Λf/8) dA ≤ 1 8ZD Ψ1(Λf)dA, and the proof of Lemma 2.4 is finished.  Proof of Theorem 2.1. 1) Consider, for every a∈D, the Berezin kernel: (2.4) Ha(z) = (1 −|a|2)2 |1−az|4· 8 One has kHakB1= 1 and kHak∞=(1 −|a|2)2 (1 −|a|)4=(1 + |a|)2 (1 −|a|)2≤4 (1 −|a|)2; hence ([7], Lemma 3.9): (2.5) kHakBΨ1≤4/h2 Ψ−1 1(4/h2),h= 1 −|a|. It follows that the function fa=1 4h2Ψ−1 1(4/h2)Hais in the unit ball of BΨ1. Now, let ξ∈Tand 0< h < 1. When z∈W(ξ, h), one has easily (see [7], proof of Theorem 4.10, with a slightly different definition of W(ξ, h)): |1−az| ≤ 5h, where a= (1 −h)ξ. It follows that then |fa(z)| ≥ (1/2500)Ψ−1 1(4/h2). Hence: 1≥ZD Ψ2|fa| kIµkdµ ≥Ψ21 2500 kIµkΨ−1 1(4/h2)µW(ξ, h), which is (2.1). 2) By Lemma 2.2, there is a constant C > 0, that we may, and shall do, assume ≥1, such that kΛfkLΨ1(µ)≤CkfkBΨ1for every f∈BΨ1. Let gbe in the unit ball of BΨ1, and apply Lemma 2.4 to f=g/C (whose norm is ≤1 yet), E=D, with hA= 1/2; we get, with ˜ C= max(1, µ(D)Ψ2(xA) + 1 8): ZD Ψ2A 64C˜ C|g|dµ ≤1 ˜ CZD Ψ2A 64C|g|dµ ≤1 ˜ Cµ(D)Ψ2(xA) + 1 8ZD Ψ1(Λf/C)dA ≤1 ˜ Chµ(D)Ψ2(xA) + 1 8i≤1, which means that kgkLΨ2(µ)≤64C˜ C/A. 2.2 Compactness Theorem 2.5 Let µbe a finite positive Borel measure on D,Ψ1and Ψ2two Orlicz functions, and let Iµ:BΨ1→LΨ2(µ)be the canonical map defined by Iµ(f) = f. One has: 1) If Iµis compact, then: (2.6) lim h→0 Ψ−1 1(1/h2) Ψ−1 21/ρµ(h)= 0. 2) In order that Iµis compact, it suffices that (2.7) lim h→0 Ψ−1 1(1/h2) Ψ−1 21/h2Kµ,2(h)= 0. 9 where c=h(0). In particular, there is a constant K=K(Ω, c)>0such that kfk2≤ KA(Ω) |f(c)|for all such functions. Moreover C > 0can be taken so that, for every positive harmonic function u: Ω →R+, one has: (3.7) 1 Cu(c)≤1 A(Ω) ZΩ u dA ≤ C u(c). Proof. By the change-of-variable formula, we have, setting F=f◦hand w=h(z): A({|f|> λ}) = ZΩ 1I{|f|>λ}(w)dA(w) = ZD 1I{|F|>λ}(z)|h′(z)|2dA(z) ≤b2ZD 1I{|F|>λ}(z)dA(z) = b2A({|F|> λ})≤b2C2 λ4|f(c)|4. We implicitly used the fact that Fmaps Dto Gand that |F(0)|=|f(c)|, so we were allowed to use the previous lemma. Moreover, we have A(Ω) = ZΩ dA(w) = ZD|h′(z)|2dA(z)≥a2A(D) = a2; so that finally: A({|f|> λ})≤C2 b2 a2 1 λ4A(Ω) |f(c)|4def =C λ4A(Ω) |f(c)|4. This ends the proof of the first part of Lemma 3.6 for Ω, with C=C2b2/a2. Finally, one has: u(c) = (u◦h)(0) = ZD (u◦h)(z)dA(z) ≥1 b2ZD uh(z)|h′(z)|2dA(z) = 1 b2ZΩ u(w)dA(w), which gives the right-hand side of (3.7), since A(Ω) ≥a2. The left-hand side is proved in the same way, since A(Ω) ≤b2. The last lemma is of a different kind. Lemma 3.7 For every analytic function f:D(z0, r)→G, one has: A({Ref > Ref(z0)/2})≥1 8K2 2AD(z0, r), where K2is the constant given by Lemma 3.5. 16 Proof. Recall Paley-Zygmund’s inequality (see [11], Proposition III.3), usually stated in probabilistic language: for any positive random variable Xon some probability space, one has, for 0< a < 1: (3.8) PX > a E(X)≥(1 −a)2[E(X)]2 E(X2), where Estands for expectation. For our problem, we will take as probability space the disk D(z0, r)equipped with the probability measure dA/r2, and as a random variable X=Ref:= u. We get from (3.8) that, since u(z0) = E(X)by the mean value property of harmonic functions: A({u > a u(z0)})≥(1 −a)2[u(z0)]2 E(u2)r2. Now, observe that √2Rew≥ |w|when w∈G; hence the function g(z) = f(z−z0)/[√2u(z0)], which maps Dinto G, satisfies |g(0)| ≤ 1and Lemma 3.5 gives E(u2)≤E(|f|2) = kfk2 2≤2K2 2[u(z0)]2. We get: A({u > a u(z0)})≥(1 −a)2 2K2 2 r2, which gives the desired result by taking a= 1/2. 3.1.2 Proof of Theorem 3.3. For technical reasons, we are going to work with functions with range in the set G. Proving Theorem 3.3 amounts to prove: Proposition 3.8 There exist constants K′>0,α1>0and λ1>1such that every analytic function f:D→Gwith |f(0)| ≤ α1satisfies, for λ≥λ1: A({|f|> λ})≤K′ λ4A({|f|>1}). It will be useful to note that √2Rew≥ |w|when w∈G. Proof. The idea of the proof is to split Fλ={|f|> λ}(actually, Fλwill be conditionned by the more regular set {Mdf > 1}) in parts on which we shall be able to apply Lemma 3.6. In order to construct these parts, we shall use a Calderón-Zygmund type decomposition adapted to the geometry of the unit disk. We are going to recall the principle of this decomposition for the convenience of the reader. Before that, we have to say that we shall begin to work not with the function fas is the statement of Proposition 3.8, but with this function multiplied by a constant (as we shall specify at the end of the proof). Nevertheless, we shall denote this new function by fyet. 17 Now, remark that for f:D→Gwith |f(0)| ≤ α < 1, one has: (3.9) |f(z)|>1 =⇒ |z|> β , for β= (1−α2)/(1+α2). In fact, setting a=f(0), the function g=f2−a2 f2+a2maps Dinto itself and vanishes at 0. By Schwarz’s lemma, |z| ≤ βimplies |g(z)| ≤ β, and since f2=a2g+a2 1−g, we get: |f(z)|2≤|a|2β+|a|2 1−β=|a|2 α2≤1. The goal of the remark (3.9) is that, in order to make the Calderón-Zygmund decomposition, we have to avoid the center 0of D, which is not covered by the sets S∈ S defined below. For convenience, we shall take β= 1/2(i.e. α= 1/√3), so we shall be only concerned by the annulus Γ = {z∈D; 1/2≤ |z|<1}. In order to perform the Calderón-Zygmund decomposition, we have to make a splitting of the annulus Γ. For that, consider the rectangle R0={x+iy ; log(1/2) ≤x < 0and 0≤y < 2π} and the family R=Sn∈NRnof all dyadic half-open rectangles, where Rnis the family of sets R={x+iy ∈C;xj≤x < xj+1 and yk≤y < yk+1}, where xj= (1 −j2−n) log(1/2) and yk= 2kπ/2n,0≤j, k ≤2n−1,n≥0. We now maps the annulus Γonto the unit disk D, using the exponential map x+iy 7→ z= ex+iy. We get a family S=Sn∈NSn. Note that the jacobian e2xof the transformation is between β2= 1/4and 1; hence the area of a set in Sn−1is less than 16 times that of a set in Sn. Recall now the Calderón-Zygmund decomposition. Consider the conditional expectation En|f|of |f|, given the σ-algebra generated by Sn, namely: (En|f|)(z) = X S∈Sn1 A(S)ZS|f|dA1IS(z). The dyadic maximal function Mdfis defined by: Mdf(z) = sup n (En|f|)(z). Note that: Mdf(z) = sup Sz 1 A(Sz)ZSz|f|dA, where the supremum is taken over all Sz∈ S containing z. 18 By Lebesgue’s differentiation Theorem (or by the martingale convergence Theorem), we know that (En|f|)(z)converges to |f(z)|,A-almost everywhere, so Mdf(z)≤1implies |f(z)| ≤ 1for almost all z∈D, and we can write: (3.10) F1={|f|>1} ⊆ {Mdf > 1}∪N , where Nis a negligible set. Hence Fλ⊆ {Mdf > 1}∪Nfor λ≥1. Now, the set {Mdf > 1}can be decomposed in a disjoint union {Mdf > 1}=FnEn, where En={z∈D; (En|f|)(z)>1and (Ej|f|)(z)≤1if j < n}. Since En|f|is constant on the sets S∈ Sn, each Encan be in its turn decomposed into a disjoint union En=FkSn,k, where Sn,k ∈ Sn. By definition, for z∈En, one has (En|f|)(z)≥1and hence 1 A(Sn,k)ZSn,k |f|dA ≥ 1for z∈Sn,k . But, on the other hand, (En−1|f|)(z)≤1, and we have, if z∈Sn,k: (En|f|)(z) = 1 A(Sn,k)ZSn,k |f|dA ≤1 A(Sn,k)ZSn−1,j |f|dA ≤ 16 1 A(Sn−1,j)ZSn−1,j |f|dA ≤ 16 , where Sn−1,j is the set of rank (n−1) containing Sn,k. Finally, reindexing the sets Sk,n, we can write {Mdf > 1}as a disjoint union (3.11) {Mdf > 1}=G l≥1 Sl, for which: (3.12) 1≤1 A(Sl)ZSl|f|dA ≤ 16 , Equations (3.10), (3.11) and (3.12) define the Calderón-Zygmund decomposition of the function f. In order to apply Lemma 3.6, we have to control (from above and from below) the values of |f|at the “center” of the sets Sl. One might think of doing this by using (3.12), but it is not always possible (for example, the function with positive real part 1/z2is not integrable on the square of vertices 0,1 + i,1−i, 2). Nevertheless, since our function is holomorphic on a domain bigger than Sl, we may enlarge Slin order to use (3.12). Let Rl∈ R be the rectangle which is mapped onto Slby the exponential, and let us round Rlin ˆ Rl, by adding half-disks, as indicated in Figure 1. Let ˆ Sl= exp( ˆ Rl). 19 Figure 1: Round set ˆ Rl 0 ^ S l l S Figure 2: Round set ˆ Sl It is essential, when using Lemma 3.6 for Ω = ˆ Sl, that the constant Cgiven by this lemma does not depend on l. This will be checked in the following way. The sets ˆ Rlcan be performed by making similarities from ˆ R0, where R0={x+iy ; log(1/2) ≤x < 0and 0≤y < 2π}. The boundary of ˆ R0is C1and it follows from [13], Theorem 3.5, for example, that ˆ R0is conformally and bi-Lipschitz equivalent to D; therefore, we are able to apply Lemma 3.6 (with c=−1 2log 2 + iπ). But if Tl(ˆ R0) = ˆ Rl, with Tl(z) = αlz+βl,αl6= 0, we have, if hl=Tl◦h,|hl′|=|αl||h′| ≤ |αl|b, as well as |hl′| ≥ |αl|a, so that aand bare respectively changed into al=|αl|aand bl= |αl|b, and the quotient bl/al=b/a remains unchanged. Now, ˆ Sl= exp( ˆ Rl)and 1/2≤ |exp(x+iy)|= ex≤1for x+iy ∈ˆ R0; hence the constant C=C2b2/a2 of Lemma 3.6 for Ω = ˆ R0becomes C≤4C2b2/a2for Ω = ˆ Sl. Therefore, one has, for every l≥1, if γlis the center (defined in an obvious way) of Sl(and hence of ˆ Sl): (3.13) A({z∈ˆ Sl;|f(z)|> λ})≤C λ4|f(γl)|4A(ˆ Sl). Now, let Dlbe the greatest open disk with center γland contained in Sl(see Figure 3). We have, by the last part of Lemma 3.6: (3.14) |f(γl)| ≤ C A(Dl)ZDl|f|dA ≤ 64C A(Sl)ZSl|f|dA ≤ 212 C . 20 S Dl l Figure 3: Disk Dl Using the fact that A(ˆ Sl)≤4A(Sl)and that Fλ⊆ {Mdf > 1} ∪ Nwhen λ≥1, we get: A(Fλ) = A(Fλ∩{Mdf > 1})≤X l≥1A({z∈ˆ Sl;|f(z)|> λ})(3.15) ≤C λ4(212C)4X l≥1A(ˆ Sl)≤252C5 λ4X l≥1A(Sl) =252C5 λ4A({Mdf > 1}). It remains to control A({Mdf > 1})by A({|f|> δ}), for some numerical δ > 0. For that, we shall use Lemma 3.7. By harmonicity and Lemma 3.6, one has, with u=Ref: u(γl)≥1/C A(ˆ Sl)Zˆ Sl u dA ≥ 1 16C 1 A(Sl)ZSl u dA ≥1 16√2C 1 A(Sl)ZSl|f|dA ≥ 1 16√2C· We now apply Lemma 3.7 and we get: A({|f|>1/64C}∩Dl)≥ A({u > 1/32√2C}∩Dl)≥ A({u > u(γl)/2}∩Dl) ≥1 8K2 2A(Dl). We obtain hence: (3.16) A({Mdf > 1}) = X l≥1A(Sl)≤16 X l≥1A(Dl)≤128K2 2A({|f|>1/64C}). The proof of Theorem 3.3 is now finished, because (3.15) and (3.16) give, for λ≥1: (3.17) A({|f|> λ})≤238 C5K2 2 λ4A({|f|>1/64C}) 21 and if fis as in the statement of Proposition 3.8, we can apply (3.17) to f1= 64C f and we get, for λ≥λ1= 64C: A({|f|> λ})≤238(64C)4C5K2 2 λ4A({|f|>1}), when |f(0)| ≤ 1/64C. 3.2 Nevanlinna counting function The Nevanlinna counting function is defined, for every analytic function ϕ:D→D, and for every w∈ϕ(D)\{ϕ(0)}, by: (3.18) Nϕ(w) = X ϕ(z)=w log 1 |z|, where each term log 1 |z|is repeated according to the multiplicity of z, and by Nϕ(w) = 0 for the other w∈D. Recall (see [9]) that, if mis the normalized Lebesgue measure on T, then the Carleson function of ϕis the Carleson function of the pull-back measure mϕof mby ϕ. We proved in [9] (Theorem 3.1 and Theorem 3.7) that the behaviour of Nϕis equivalent to that of the Carleson function ρϕin the following way. Theorem 3.9 ([9]) There exist two universal constants C, c > 1, such that, for every analytic self-map ϕ:D→D, one has: (3.19) sup w∈W(ξ,h)∩D Nϕ(w)≤C mϕ[W(ξ, c h)] , and (3.20) mϕW(ξ, h)≤C1 AW(ξ, ch)ZW(ξ,ch) Nϕ(z)dA(z) for 0< h < 1small enough. We are going to deduce from Theorem 3.9 the same result in the 2-dimensional case. The Nevanlinna counting function of order 2is defined (see [15], § 6.2), for w∈D\{ϕ(0)}, by: (3.21) Nϕ,2(w) = X ϕ(z)=wlog(1/|z|)2, where each preimage zof wappears as often as its multiplicity. The partial Nevanlinna counting function is defined, for 0< r ≤1by: Nϕ(r, w) = X ϕ(z)=w,|z|<r log(r/|z|) 22 and we have ([15], Proposition 6.6, where a misprint occurs): (3.22) Nϕ,2(w) = 2 Z1 0 Nϕ(r, w)dr r· One has: Theorem 3.10 There exists a universal constant C > 1, such that, for every analytic self-map ϕ:D→D, one has: (3.23) (1/C)ρϕ,2(h/C)≤sup |w|≥1−h Nϕ,2(w)≤C ρϕ,2(C h), for 0< h < 1small enough. Proof. Let w0=ϕ(0) and set: u(z) = w0−ϕ(z) 1−w0ϕ(z)· Since u(0) = 0, Schwarz’s lemma gives |u(z)| ≤ |z|. Hence there is no zwith |z|< t such that ϕ(z) = wwhen t≤ |w0−w|/|1−w0w|:= |u0(w)|. It follows that (3.22) actually writes: (3.24) Nϕ,2(w) = 2 Z1 |u0(w)| Nϕ(r, w)dr r· 1) It follows from (3.24) that: Nϕ,2(w)≤2 |u0(w)|2Z1 0 Nϕ(r, w)r dr . But (see [9], Lemma 3.4) 1/|u0(w)|=|w−w0|/|1−ww0|>1/3when 1−|w|< (1 −|w0|)/4; therefore, for 1−|w|<(1 −|w0|)/4, we have: Nϕ,2(w)≤18 Z1 0 Nϕ(r, w)r dr . Now, Nϕ(r, w) = Nϕr(w), where ϕr(z) = ϕ(rz), and it follows from (3.19) that, for w∈W(ξ, h), with ξ∈Tand h > 0small enough, one has: Nϕ,2(w)≤18 Z1 0 Cmϕr[W(ξ, c h)] r dr = 18CZ1 0 m{eiθ ;ϕ(reiθ)∈W(ξ, c h)}r dr = 9CA{z∈D;ϕ(z)∈W(ξ, c h)}. 2) Conversely, it follows from (3.22) that: Nϕ,2(w)≥2Z1 0 Nϕ(r, w)r dr ; 23 hence: 1 A[W(ξ, ch)] ZW(ξ,ch) Nϕ,2(z)dA(z) ≥2Z1 01 A[W(ξ, ch)] ZW(ξ,ch) Nϕ(r, z)dA(z)r dr ≥2 CZ1 0 mϕr[W(ξ, h)] r dr =2 CZ1 0 m{eiθ ;ϕ(reiθ)∈W(ξ, h)}r dr =1 CA{z∈D;ϕ(z)∈W(ξ, h)}, and that finishes the proof of Theorem 3.10.  Remark. Actually, the proof shows that for some constant C > 1, one has: (3.25) (1/C)Aϕ[W(ξ, h/C)] ≤sup w∈W(ξ,h) Nϕ,2(w)≤CAϕ[W(ξ, Ch)] , for every ξ∈Tand 0< h < 1small enough. Since the ℓ2-norm is less than the ℓ1-norm, one has Nϕ,2(w)≤[Nϕ(w)]2, it follows hence from Theorem 3.9 and Theorem 3.10 (actually (3.25)) that, for some constant C > 1, one has: (3.26) A({z∈D;ϕ(z)∈W(ξ, h)})≤Cm({u∈T;ϕ∗(u)∈W(ξ, Ch)})2, for every ξ∈Tand every 0< h < 1small enough, a fact which does not seem easy to proved in a straightforward way. In particular, for some constant C > 0, one has, for h > 0small enough: ρϕ,2(h)≤C[ρϕ(Ch)]2. Corollary 3.11 For every analytic self-map ϕ:D→Dand for every Orlicz function Ψ, the composition operator Cϕ:BΨ→BΨis compact if and only if: (3.27) lim h→0 Ψ−1(1/h2) Ψ−11/νϕ,2(h)= 0 , where νϕ,2(h) = sup|w|≥1−hNϕ,2(w). Proof. If Cϕis compact, Theorem 3.2 gives, for every A > 0, an hA>0such that, for 0< h ≤hA: Ψ−1(1/h2)≤1 AΨ−11/ρϕ,2(h). Then Theorem 3.10 gives: νϕ,2(h)≤C ρϕ,2(Ch)≤C/Ψ[AΨ−1(1/C2h2)] , 24 i.e. AΨ−1(1/C2h2)≤Ψ−1[C/νϕ,2(h)]; and then, by concavity, since C > 1: 1 C2Ψ−1(1/h2)≤Ψ−1(1/C2h2)≤1 AΨ−1C/νϕ,2(h)≤C AΨ−11/νϕ,2(h), which implies (3.27). The converse follows the same lines.  Corollary 3.12 The compactness of the composition operator Cϕ:BΨ→BΨ implies that: lim |z|→1 Ψ−11/[1 −|ϕ(z)|]2 Ψ−11/(1 −|z|)2= 0 . This corollary was proved in [7], Theorem 5.7, by a more direct method; and we also showed that the condition is sufficient when Ψgrows fast enough (namely, satisfies the condition ∆2). However, we do not know whether it is not sufficient for some symbol ϕand some Orlicz function Ψ(see the remark at the end of the paper). Nevertheless, it follows easily from Corollary 3.11 that this condition is sufficient when ϕis finitely-valent (see [9], proof of Theorem 5.3). Proof. Since Nϕ,2ϕ(z)≥log(1/|z|)2≥(1 − |z|)2, it follows from Corollary 3.11 that, for every A > 0, one has: (1 −|z|)2≤1 ΨAΨ−11/(1 −|ϕ(z)|)2; that is: Ψ−11/(1 −|ϕ(z)|)2 Ψ−11/(1 −|z|)2≤1/A , and that proves Corollary 3.12.  4 Comparison of the compactness of composition operators on Hardy-Orlicz spaces and on Bergman-Orlicz spaces In the classical case (Ψ(x) = xp,1≤p < ∞), it is known ([12], Theorem 3.5, with Proposition 2.7) that the compactness of Cϕ:Hp→Hpimplies the compactness of Cϕ:Bp→Bp. On the other hand, we implicitly proved in [7], Theorem 5.7, that when Ψgrows very fast (namely, satisfies the so-called ∆2condition), then the compactness of Cϕ:HΨ→HΨimplies the compactness of Cϕ:BΨ→BΨ. Let us write why: it is easy to show (see [9], proof of Theorem 4.3) that the compactness of Cϕ:HΨ→HΨimplies that Ψ−1[1/1−|ϕ(z)|)]/Ψ−1[1/(1 −|z|)] tends to 0as |z|goes to 1, and we actually proved in [7], Theorem 5.7, that, when Ψ∈∆2, this last condition implies the compactness of Cϕ:BΨ→BΨ. The next proposition gives a condition on Ψ which, though not very satisfactory, includes the cases Ψ(x) = xpand Ψ∈∆2 and for which compactness on HΨimplies compactness on BΨ. 25 Pascal Lefèvre, Univ Lille Nord de France F-59 000 LILLE, FRANCE UArtois, Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, F-62 300 LENS, FRANCE pascal.lefevr[email protected]r Daniel Li, Univ Lille Nord de France F-59 000 LILLE, FRANCE UArtois, Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, Faculté des Sciences Jean Perrin, Rue Jean Souvraz, S.P. 18, F-62 300 LENS, FRANCE daniel.[email protected] Hervé Queffélec, Univ Lille Nord de France F-59 000 LILLE, FRANCE USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524, F-59 655 VILLENEUVE D’ASCQ Cedex, FRANCE [email protected]le1.fr Luis Rodríguez-Piazza, Universidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático, Apartado de Correos 1160, 41 080 SEVILLA, SPAIN [email protected] 32